Example: No Interaction, interpret Main Effects
Treatment Structure: 3 x 2 Full Factorial
Design Structure: CRD with r = 2 stores per treatment combination.
Response: Bread Sales
\[y_{ijk}=\mu+\alpha_i+\beta_j+\alpha\beta_{ij}+\epsilon_{ijk} \text{ with } \epsilon_{ijk} \sim \text{ iid }N(0,\sigma^2)\]
\[\text{for } i=1,2,3,…,a; j=1,2,…,b; k=1,2,….,r\]
Interaction
\[H_0:\text{ All } \alpha\beta_{ij} = 0 \text{ vs } H_A: \text{At least one } \alpha\beta_{ij} \ne 0\]
Main Effect of A
\[H_0:\text{ All } \alpha_{i} = 0 \text{ vs } H_A: \text{At least one } \alpha_{i} \ne 0\]
Main Effect of B
\[H_0:\text{ All } \beta_{j} = 0 \text{ vs } H_A: \text{At least one } \beta_{j} \ne 0\]
Analysis of Variance Table
Response: sales
Df Sum Sq Mean Sq F value Pr(>F)
height 2 1544 772.00 74.7097 5.754e-05 ***
width 1 12 12.00 1.1613 0.3226
height:width 2 24 12.00 1.1613 0.3747
Residuals 6 62 10.33
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Inspecting the interaction plot between height and width on bread sales, does there visually appear to be a significant interaction?
What are the marginal mean sales for each height (sig main effect)? Which heights differ?
NOTE: Results may be misleading due to involvement in interactions
height emmean SE df lower.CL upper.CL
bottom 44 1.61 6 40.1 47.9
middle 67 1.61 6 63.1 70.9
top 42 1.61 6 38.1 45.9
Results are averaged over the levels of: width
Confidence level used: 0.95
contrast estimate SE df lower.CL upper.CL t.ratio p.value
bottom - middle -23 2.27 6 -29.97 -16.03 -10.119 0.0001
bottom - top 2 2.27 6 -4.97 8.97 0.880 0.6714
middle - top 25 2.27 6 18.03 31.97 10.999 <0.0001
Results are averaged over the levels of: width
Confidence level used: 0.95
Conf-level adjustment: tukey method for comparing a family of 3 estimates
P value adjustment: tukey method for comparing a family of 3 estimates
height emmean SE df lower.CL upper.CL .group
top 42 1.61 6 36.7 47.3 A
bottom 44 1.61 6 38.7 49.3 A
middle 67 1.61 6 61.7 72.3 B
Results are averaged over the levels of: width
Confidence level used: 0.95
Conf-level adjustment: sidak method for 3 estimates
P value adjustment: tukey method for comparing a family of 3 estimates
significance level used: alpha = 0.05
NOTE: If two or more means share the same grouping symbol,
then we cannot show them to be different.
But we also did not show them to be the same.